Classical stochastic systems with fast-switching environments: Reduced master equations, their interpretation, and limits of validity.

We study classical Markovian stochastic systems with discrete states, coupled to randomly switching external environments. For fast environmental processes we derive reduced dynamics for the system itself, focusing on corrections to the adiabatic limit of infinite timescale separation. We show that this can lead to master equations with bursting events. Negative transition rates can result in the reduced master equation, leading to unphysical short-time behavior. However, the reduced master equation can describe stationary states better than a leading-order adiabatic calculation, similar to what is known for Kramers-Moyal expansions in the context of the Pawula theorem [R. F. Pawula, Phys. Rev. 162, 186 (1967)PHRVAO0031-899X10.1103/PhysRev.162.186; H. Risken and H. Vollmer, Z. Phys. B 35, 313 (1979)ZPBBDJ0340-224X10.1007/BF01319854]. We provide an interpretation of the reduced dynamics in discrete time and a criterion for the occurrence of negative rates for systems with two environmental states.